I would have done it like this - it's not as elegant as Galactus's approach, but may be a bit easier to follow:
You know that the first coefficient is a^n = 729. So consider what are the square roots, cube roots, etc of 729 that are integers? If you convert 729 to is prime factors you see that 729 = 3^6. So either:
729 = 3^6 (a = 3 and n= 6), or
729 = 9^3 (a = 9, n = 3), or
729 = 27^2 (a = 27, n = 2)
The second coefficient is n*a^(n-1)*b = 2916, so let's see if any of the combinations of a and n work:
For a = 3, n=6:
6*3^5*b = 2916, or b = 2
For a = 9, n = 3:
3*9^2*b = 2916, or b = 12
For a = 27, n = 2:
2*27*b = 2916, or b = 54
Now try these combinations of a, b, and n into the equation for the third coefficient, which you know equals 4860, and see if they check out. The equation of the third coefficient is:
n*(n-1)/2 * a^(n-2) * b^2 = 4860.
For a = 3, b =2, n = 6:
6*5/2 * 3^4 *2^2 = 4860, so this checks out.
For a = 9, b = 12, n=3:
3*2/2 *9*12^2 = 3888, so this is not correct.
For a = 27, b = 54, n = 2:
2*1/2 * 27^0 *54^2 = 2916, so this is also not correct.
Hence the only answer is: a = 3, b =2, and n = 6.
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